Suppose that the weight of navel oranges is normally...

Suppose that the weight of navel oranges is normally distributed with mean μ = 8 ounces and standard deviation σ = 1.5 ounces. We can write X ∼ N(8, 1.5). Answer the following questions: What proportion of oranges weigh more than 11.5 ounces? What proportion of oranges weigh between 6.2 and 7 ounces? Draw and label the density curves (f(x) and Φ(z)) and shade the areas that will help you solve the problem.

Transcript

00:01 This question is given the weights of naval oranges is normally distributed with the mean of 8 ounces and standard deviation 1 .5 ounces.

00:12 So the x is the weight of a naval orange.

00:15 So x follows the normal distribution.

00:17 Mean is 8.

00:18 Standard deviation is 1 .5.

00:20 So variance will be 1 .5 square.

00:23 Now, proportion of oranges that weigh more than 11 .5 ounces.

00:29 So we're looking at probability of x greater than 11 .5.

00:36 Now let's draw the normal distribution.

00:41 So it will look like this.

00:44 So this will be for the normal distribution of fx, x follows the normal distribution.

00:50 The mean is it, variance is 1 .5 square.

00:59 The normal distribution is symmetrical about the center and that's the mean.

01:07 And that's that would be 8.

01:10 Now we're looking at greater than 11 .5.

01:13 So 11 .5 is over here.

01:16 So greater than 11 .5, we're looking at this part here, this shaded part here, the area under the normal curve here.

01:23 Now, we want to do standard normal.

01:26 So we know that standard normal is z, and there is x minus the mean, divided by standard deviation.

01:36 So what i'm going to do is i'm going to apply an x minus the mean, which is on both sides and then divide by the standard deviation which is 1 .5.

01:49 So now this part will become your standard normal which is your z and over here this would be 7 over 3...

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